All questions with tag [math: algebraic-combinatorics]


Given a list of $2^n$ nonzero vectors in $GF(2^n)$, do some $2^{n-1}$ of them sum to 0?

Let $G=(\mathbb{Z/2Z})^n$ written additively, $n>1$. (you can think of it as $\mathbb{F}_{2^n}$ but I didn't find that useful... yet) Let $v_i$ be nonzero elements of $G$ for $i \in \{1 \dots 2^n \}$ so that $\sum_{i=1}^{2^n} v_i=0$. Is it possible to find a subset $J \subset \{1 \dots 2^n \}$, $|J|=2^{n-1}$ so that $$\sum_{j \in J} v_j=0...
2022-07-08 17:32:28

Identity involving partitions of even and odd parts.

First, represent by $p_E(n)$ be the variety of dividings of $n$ with an also variety of components, and also allow $p_O(n)$ be those with a weird variety of components. In addition, allow $p_{DO}(x)$ be the variety of dividings of $n$ whose components stand out and also weird. Ultimately, allow $c(n)$ be the variety of dividings of $n$ which are...
2022-07-01 15:04:20

Uses of Chevalley-Warning

In the current IMC 2011, the last trouble of the 1st day (no. 5, the hardest of that day) was as adheres to: We have $4n-1$ vectors in $F_2^{2n-1}$: $\{v_i\}_{i=1}^{4n-1}$. The trouble asks: Confirm the presence of a part $A \subseteq [4n-1], |A| = 2n$ such that $\sum_{i \in A} v_i = \vec{0}$. Exists a remedy that makes use of the Chevalley-Wa...
2022-06-08 22:05:34